Smooth displaced mesh
Since the displacement modifier uses textures, you can make use of texture sampling options which apply with the example you give.

In the texture's Image Sampling panel, make sure Interpolation is enabled, and increase the filter size (values between [1.5 ... 2.0] give quite smooth result, 3.0 and above are smooth but you loose definition).

You may also want to change the color-space of the image between linear/srgb.